2.3 Proofs
2.3.1 Proof of Theorem 2.1.1
In this subsection we first present the proof of Theorem 2.1.1 for the case when the envi- ronment is started from all sites occupied. Essentially the same proof can be applied to the case where the environment is started from all sites vacant. We comment at the end of this subsection on which changes to the proof are necessary for this case.
The main idea is to show thatρt(ξ)is sub-additive, by using thatξhas an attractive graphical
representation coupling and Lemma 2.2.1. Subsequently, the subadditive ergodic theorem applied toρt(ξ)yields that t−1ρt(ξ)converges towards a deterministic constant. This, in
turn, identifies the limiting speed.
Let ξ be an IPS with an attractive graphical representation coupling, Pb, where byI we
denote the corresponding collection of Poisson point processes. In order to formulate the proof, we have to be more specific aboutI and writeI as a countable set of Poisson point processes indexed by the latticeZd,I = (Xy1)y∈Zd,(X
2
y)y∈Zd, . . .
, where everyXi y is an
(independent) Poisson point process on[0,∞). Further, forx∈Zdandt∈[0,∞)letΘx,t
be the space-time shift operator on the realisations ofI: Θx,t (Xs,y1 ),(X 2 s,y), . . . s∈[0,∞),y∈Zd= (X 1 s+t,y+x),(X 2 s+t,y+x), . . . s∈[0,∞),y∈Zd.
For the contact process, the setIas considered in Section 2.4, consists of the Poisson pro- cessesHx, Ix,e:x, e∈Zd,|e|= 1 , andΘx,tshifts crosses and arrows in space byxand
in time byt. Further, in [55, Theorem 2.5],Iis the set of birth and death events (which in [55] are denoted byTy,i
n :n≥0, y∈Zd, i∈ {0,1} ).
To emphasise the graphical representation, we writeρt(ξ) =ρt(η,I, N, O, V)forξ0 = η and letbPdenote the joint law of the graphical construction coupling andN, OandV. Note
that, by Lemma 2.2.1 and (2.1.4), for anyη ∈Ω,
ρt(η,I, N, O, V)≤ρt(¯1,I, N, O, V), Pb−a.s. (2.3.1)
Moreover, let
N(s)= (Nt(s))t≥0:= (Nt+s−Ns)t≥0,
O(s)= (On(s))n≥0:= (On+γρs(ξ))n≥0,
V(s)= (Vn(s))n≥0:= (Vn+Ns−γρs(ξ))n≥0.
Similar toΘx,twe introduce the space-time shiftθx,tonΩ[0,∞)by
with space-shiftθxintroduced in (2.1.1). Next, define the continuous-time process(Xt,s)0≤t≤s
by
Xt,s:=ρs−t(¯1,ΘWt,tI, N
(t), O(t), V(t)), for0≤t≤s. Note that, ifξis such thatξ0= ¯1, then
X0,s=ρs(¯1,I, N, O, V) =ρs(ξ), s∈[0,∞).
In the next statement and in the proceedings, forµ∈ P(Ω)andx∈ Zd, we writebPµ,xto
emphasise the starting configuration of bothξand(Wt).
Lemma 2.3.1(Sub-additivity). The process(Xt,s)0≤t≤shas the following properties.
i) X0,0= 0and for allt, s∈[0,∞):X0,t+s≤X0,t+Xt,t+s.
ii) For allt∈(0,∞),(Xt,k+t)k≥1has the same distribution as(X0,k)k≥1.
iii) For allt∈[0,∞),(X(k−1)t,kt)k≥1is a sequence of i.i.d. random variables.
iv) For allt∈[0,∞), the expectationEbδ1¯,o[X0,t]is finite andX0,t≥0.
Proof. Fixt, s ∈ [0,∞)and recall (2.3.1). By the Markov property of the Poisson point processI, we have that
X0,t+s=ρt+s(¯1,I, N, O, V) =ρt(¯1,I, N, V, O) +ρ(s, θWt,tξ,ΘWt,tI, N (t), O(t), V(t)) ≤ρt(¯1,I, N, O, V) +ρ(s,¯1,ΘWt,tI, N (t), O(t), V(t)) =X0,t+Xt,t+s.
Properties i) and ii) follow from the equalityX0,0 = 0, the translation invariance in (2.1.1) and the equality in distributionX0,s =Xt,t+s. Moreover, iii) follows by the Markov prop-
erty ofξand the graphical representation. Lastly, property iv) holds trivially, sinceX0,t is
non-negative by definition and sinceX0,t≤Nt.
Lemma 2.3.1 enables us to prove the SLLN for the processρt(ξ)whenξis initialised at time
0by¯1, by applying the subadditive ergodic theorem.
Theorem 2.3.2(Law of large numbers forρt(ξ)). Assume thatξhas an attractive graphical
representation coupling. There existsρ1∈[0,1]such that
lim t→∞ 1 tρt(ξ) =ρ1 bPδ¯1,o-a.s. and inL 1. (2.3.2) Moreover,ρ1= inft≥1t−1Ebδ¯1,o(ρt(ξ)).
2.3 Proofs
Proof. By Lemma 2.3.1 we know thatX satisfies property a)-d) of [82, Theorem VII.2.6]. In particular, by the independence property in Lemma 2.3.1iii), the process is stationary and ergodic. Hence, the conclusion of Theorem 2.3.2 holds whenttakes integer values. This can easily be extended to continuoust. Indeed, for anyt∈(0,∞)we have that
X0,btc ≤X0,t≤X0,dte. (2.3.3)
In particular, by dividing bytin (2.3.3) and takingt→ ∞(as in (2.3.2)), we conclude the proof.
We are now in position to present the proof of Theorem 2.1.1.
Proof of Theorem 2.1.1. By the construction in Section 2.2.1,Wtcan be written as
Wt= ρ(Nt,ξ) X i=1 ∞ X m=1 1[p1(m−1),p1(m))(Oi)zm ! + Nt−ρ(Nt,ξ) X j=1 ∞ X n=1 1[p0(n−1),p0(n))(Vj)zn ! . Dividing byt >0gives Wt t = ρ(Nt, ξ) t 1 ρ(Nt, ξ) ρ(Nt,ξ) X i=1 ∞ X m=1 1[p1(m−1),p1(m))(Oi)zm ! +Nt−ρ(Nt, ξ) t 1 Nt−ρ(Nt, ξ) Nt−ρ(Nt,ξ) X j=1 ∞ X n=1 1[p0(n−1),p0(n))(Vj)zn ! .
Taking the limit ast→ ∞and applying Theorem 2.3.2 we obtain lim
t→∞
1
tWt=ρ1u1+ (1−ρ1)u0 Pδ¯1,o-a.s. and inL
1,
whereρ1 is as in Theorem 2.3.2 andu0, u1 ∈ Rd are as in (2.1.3). This proves Theorem
2.1.1 for the case when the environment is started from all sites equal to1.
We next comment on the changes necessary in the argument for proving Theorem 2.1.1 when started from all sites equal to0. For this case we can define the process(Yt,s)0≤t≤sgiven by
Yt,s:=ρ(s−t,¯0,ΘWt,tI, N
(t), O(t), V(t))for0≤t≤s.
By the same arguments as in Lemma 2.3.1 we can prove that−Y is a sub-additive process satisfying property ii) and iii) as in Lemma 2.3.1. Moreover, sinceY0,tis dominated byNt
it follows thatEbδ¯0,o[Y0,t]≤Ebδ¯0,o[Nt] =t. This is sufficient in order to apply [82, Theorem
VII.2.6]. By a literal adaptation of the proof underbPδ¯1,oabove we obtain
lim
t→∞
1
tWt=ρ0u1+ (1−ρ0)u0 Pδ¯0,o-a.s. and inL
1,
whereρ0is the limit in Theorem 2.3.2 whenPbδ¯1,ois replaced bybPδ¯0,o. This completes the